exploring infinity : number sequences in modern art
Götz Pfander and Isabel Wünsche
41
42
exploring infinity : number sequences in modern art
Götz Pfander & Isabel Wünsche
43
44
exploring infinity : number sequences in modern art
Götz Pfander & Isabel Wünsche
45
46
exploring infinity : number sequences in modern art
Götz Pfander & Isabel Wünsche
47
48
exploring infinity : number sequences in modern art
Götz Pfander & Isabel Wünsche
49
Illustrations :
page 42
Roman Opalka, OPALKA 1965/ 1-∞, 1965, Detail 774 509 – 800 148
Image from : Karin von Maur, Magie der Zahl in der Kunst des 20. Jahrhunderts, Stuttgart,
1997, Ill. 7.47 and 7.47a, p. 175.
page 43
Rune Mields, Das Sieb des Eratosthenes III (The Sieve of Eratosthenes III), 1977
Image from : http://home.ph-freiburg.de/grevsmuehlfr/material/forschung/22%20F%E4cherverbindende%20Studien/MT%20124%20-%20NUMBER%20IN
VESTIGATIONS.pdf
page 44
Mario Merz, Pythagoras’ Haus, 1994
Image from : Karin von Maur, Magie der Zahl in der Kunst des 20. Jahrhunderts, Stuttgart,
1997, Ill. 9.4, p. 205.
page 45
Michael Müller, Ulam Rot/Grün (Ulam Red/Green), 2005
Image from the artist.
page 46
Michael Müller, 47 48, 2005
Image from the artist.
page 47
Michael Müller, Ulam Spirale (Ulam Spiral), 2006
Image from the artist.
page 48
Michael Müller, Ulam Ringe (Ulam Rings), 2005
Image from the artist.
page 49
Michael Müller, Ulam Gebirgszug (Ulam Mountain Range), 2005
Image from the artist.
50
exploring infinity : number sequences in modern art
A fascination with numbers has shaped the work of numerous
artists throughout the centuries. Whereas in earlier times artists focused
on the mystical symbolism of individual numbers and their perceived
capacity to give structure to the complexity of the universe, modern artists
have concentrated more on an analysis of the pictorial language and the
conceptual dimension of numbers. Their artistic approaches vary greatly
in respect to both form and content and often counter the rational thought
structures of our age with alternative ways of thinking and seeing. Numbers and numerical systems play an important role in the work of artists
such as Hanne Darboven, Charles Demuth, Jasper Johns, Joseph Kossuth,
Mario Merz, Rune Mields, Michael Müller, On Kawara, Roman Opalka,
and others [1]. As Karin Maur pointed out in a recent exhibtion catalogue,
these artists use numbers as signs of modernity, found objects and random
subjects. They explore numerical writing and the relationships between
numbers and languages, and investigate the development from image to
arithmetical progression, codes of war, and alogical calculation. They reflect
upon the role of numbers in nature, human proportions, and everyday life
and are fascinated by the meter of time and metaphysical expression of
infinity as well as numerical mysticism and the harmonic orders [2].
Since the 1960s, conceptual artists such as Hanne Darboven, On
Kawara, and others have explored the significance of numbers as universal
language and measurement as well as the interrelationships of numbers
and their role within a larger ordering system. Whether simply written out
or used as graphic elements, numbers have served as an artistic medium
to carry specific messages, to visualize complex systems, and to convey
metaphysical meanings.
In this essay, we focus on another aspect of numbers in art, namely,
the explicit use of various natural number sequences. In particular, we
look at the role of the natural numbers in the work of Roman Opalka, the
prime number sequence in the work of Rune Mields, the Fibonacci number
sequence often used by Mario Merz, and the U-sequence in the work of
Michael Müller. Of interest are not only the wider socio-cultural contexts
and larger philosophical meanings behind the use of these sequences by
contemporary artists, but also the mathematical theory underlying the
respective sequences, which we will explore in detail.
The mathematics of the natural numbers and sequences of natural
numbers such as the Fibonacci numbers are studied within the mathematical
field of number theory [3]. Even though there exist uncountably many
sequences of natural numbers, that is, the set of all sequences cannot be
Götz Pfander & Isabel Wünsche
51
enumerated, those sequences that have mostly been studied in mathematics
are identified with a number in Neil Sloane’s The online encyclopedia of integer
sequences [4].i The numbering of Sloane’s encyclopedia is used in this essay,
for example, the sequence of even numbers is referred to as [A005843].
Roman Opalka : natural numbers and the passage of time
The Polish artist Roman Opalka (*1931) uses the sequence of natural
numbers [A001477]
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, … .
to visualize the passage of time. In 1965, he began his lifelong task of
painting numbers in sequence, beginning with 1, then 2, going on, in theory,
to infinity. Accordingly, the name of his work is : OPALKA 1965/1 - ∞
(illustration 1). It is an ongoing work, a continuous process that will never
be completed and only end when the artist dies. Traditionally, artists have
used symbols such as clocks or the human skull to allegorically represent
the passing of time and the fleetingness of life and life’s vanities. Opalka
uses instead the natural numbers to visualize the passing of time. Purposely
stripped of all manner of expression, the paintings constitute a record of
the artist counting from the number one towards infinity. Each work in the
series, entitled Detail, is identical in size (1.96 x 1.35 m) and is covered with
numbers painted in white, in level rows, from edge to edge. The first painting
in the series includes the numbers 1 to 35328; the second panel then begins
with 35329.ii The artist brightens the originally black canvas by 1% from
painting to painting, so that the second canvas is 1% brighter than the first,
the third canvas is 1% brighter than the second and 1.99% brighter than
the first, the fourth canvas is 2.97% brighter than the first, and so on. The
« last » canvas in the series will consist of white numerals on an almost white
background, again emphasizing the continuity of his work and the passage
of time. In the tradition of Kazimir Malevich’s White on White painting of
1918, Opalka is continually approaching infinity with his series of numbers
and ever-lighter canvases, but will never reach it. The paintings, an ongoing
numerical record of the passage of time, are accompanied by a recording of
the artist counting aloud in his native language and by photographs of the
artist taken at regular intervals as he works on the canvases. Having spent
more than half of his life on this single work, Opalka has meanwhile passed
the number 5000000 ; there are now more than 200 panels in the series.
i
ii
52
In fact, there are as many sequences of natural numbers as there are numbers on the number line.
Putting Opalka’s first detail under scrutiny, one can find some mistakes in his attempt to write out all natural numbers in
increasing order. For example, the numbers 34009, 33010, 33011, 34015, 34016 are listed successively.
exploring infinity : number sequences in modern art
Rune Mields : chaos and order in the prime numbers
Prime numbers are positive integers larger than 1 that are only divisible
by 1 and by themselves. Numbers which are not prime and which therefore
have nontrivial divisors are called composite numbers. The sequence of
primes starts with
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, … .
The Greek philosopher and mathematician Euclid (circa 325 – 265 B. C.)
showed that the sequence of primes is infinite, just like the sequence of
natural numbers [5]. In fact, if there were only finitely many prime numbers, then every natural number would be divisible by at least one of those
finitely many prime numbers. Therefore, the number computed by adding 1
to the product of all these finitely many prime numbers will be divisible by
at least one of these prime numbers. Since the product itself is also divisible
by that same prime number, the difference, namely 1, must be divisible
by that prime number as well. This is obviously nonsense and, therefore,
the assumption that there are only finitely many primes must have been
incorrect in the first place. Hence, there are infinitely many prime numbers.
This classical proof by contradiction helps neither to find infinitely
many prime numbers, nor to determine whether a given number is prime.
Indeed, it is simple to show that 229 is prime, in fact, it is the 50th prime, but a
significant effort is needed to check whether a number with several hundred
or even several thousand digits is prime. Even though prime numbers with
thousands of digits are not used for counting – some scientists estimate that
an integer with 130 digits should be sufficient to represent the number of
electrons in the observable universe – large primes are in fact useful and play
a commercial role in the encryption of information [6].
To find fast algorithms which determine whether a large number is
prime remains an active research topic in number theory. A recent and
major theoretical achievement in this direction was the justification that the
amount of computations needed to see whether an N-digit number is prime
does not grow exponentially with N, but can be bounded by C· N12 (C is a
large constant that does not depend on the number of digits N) [7].
Today, hundreds of volunteers donate computer resources to the Great
Internet Mersenne Prime Search (GIMPS) project which coordinates the hunt for
larger and larger primes [8]. The GIMPS project focuses on the discovery of
the so-called Mersenne primes, that is, of primes of the form 2n-1, where n
is prime. Their effort delivers a new largest known prime approximately twice
a year. The current record is 232582657-1. It was discovered on September 6th,
Götz Pfander & Isabel Wünsche
53
2006, and its 9’808’358 digits barely missed the 10 million digits necessary to
collect the prize of $ 100’000 for cooperative computing from the Electronic
Frontier Foundation [9]. The inclusion of this currently largest known prime,
digit by digit, would require approximately an extra 3’800 pages of this issue
of Arkhaï. This essay alone would consequently be almost 40 centimeters
thick.
The quest to find larger and larger prime numbers has occupied
mathematicians and artists alike. The Cologne-based artist Rune Mields
(*1935) has concentrated in her work on exploring the largest known prime
numbers for decades. In the 1970s, she created a series of works visualizing
the primes between 1 and 120000 in the Chinese-Japanese Sanju system.
When, in 1980, the then largest prime number 2444497-1, a number with 13’396
digits, was discovered, she used a computer printout of this prime number to
identify the digits 0 and 9 within this prime, as well as their frequency. After
a new large prime number was discovered by the German mathematician
Wilfried Keller in 1984, Mields used a printout of this prime for her work
Die Söhne der Mathematik (The Sons of Mathematics, 1988) [10].
More generally, Mields focuses on the contrast between chaos and
order by exploring old and new ordering systems in their historical and
cultural continuity. She has worked with 96 known number systems and
sequences, among them stick counting, the Fibonacci series, the Sanju prime
numbers, the magical square, and the 6 x 12 number-picture system. In her
study of prime numbers, Mields has also focused on visualizing repunit
primes – primes whose digital presentation is a string of 1s, the only known
repunits have 2 digits (11), 13 digits (1111111111111111111), 19 digits, 317
digits and 1’031 digits – and the so-called twin primes 3,5 ; 5,7 ; 11,13 ; 17,19 ;
29,31 ; … Her interest in twin primes touches upon a famous open problem
in number theory, namely the Twin Prime Conjecture which states that there are
infinitely many prime numbers p with the property that p+2 is also a prime number [11].
This problem was already mentioned by Euclid in his Elements. The largest
known example for twin primes is the pair
100314512544015· 2171960-1, 100314512544015· 2171960+1,
each of which has 51’780 digits. Mields explores these mathematical
questions from an artist’s point of view and accompanies mathematicians as
they attempt to proceed into uncharted territory of number theory.
Mields also represents prime numbers in various visual forms in
an effort to reveal structure and provide insight. In her series Das Sieb
des Eratosthenes III (The Sieve of Eratosthenes III, 1977, illustration 2),
54
exploring infinity : number sequences in modern art
she applies a famous procedure, named after the Alexandrian scholar
Eratosthenes of the 3rd century B. C. and still taught in schools today,
for finding all prime numbers smaller than a given large number. She
summarizes this procedure as follows [12] :
1. 2 is a prime number so put a circle round the 2 on your square.
2. Now cross out all the other multiples of 2. If you can spot a pattern you
will be able to do this very quickly.
3. Now put a circle round the next number after 2 that isn’t crossed out,
that is 3.
4. Now cross out all the other multiples of 3. You will find that some of
them are crossed out already. Once again, look for a pattern.
5. Now move on to the 5. Circle it and cross out all its other multiples. Keep
going… all the circled numbers will be prime ! !
While the hunt for bigger and bigger prime numbers necessitates the
development of fast algorithms and fast computers, there are many open
questions analogous to the Twin Prime Conjecture waiting for a « pencil on
paper » proof. The Riemann Hypothesis [13], likely the most sought after
question in mathematics since Andrew Wiles was able to confirm Fermat’s
Last Theorem in 1995, is one of seven open problems in mathematics whose
corroboration carries a prize of $1’000’000 from the Clay Mathematics
Institute [14]. The Riemann Hypothesis discusses the distribution of primes
among integers. As was shown above, there are infinitely many prime
numbers. Still, one can ask what the proportion of prime numbers to
composite numbers is, or, loosely speaking, what is the probability that a
randomly chosen natural number is prime. The Riemann Hypothesis states
a more precise estimate of the proportion of primes to composites than
the long established Prime Number Theorem, which asserts that there are
approximately n/log(n) prime numbers smaller than any given number n.
This implies that large numbers are less likely to be prime numbers than
small numbers, something which becomes intuitively clear through, for
example, The Sieve of Eratosthenes. The Prime Number Theorem’s accuracy is
quite impressive. For example, n/log(n) estimates 21 prime numbers smaller
than 100, 144 smaller than 1000, 1’085 smaller than 10000, and 8’685 smaller
than 100000, while the precise numbers are 25 primes smaller than 100,
168 smaller than 1000, 1’229 smaller than 10000, and 9’592 smaller than
100000.
The equally famous Goldbach Conjecture from 1742 states that every
even integer larger than two can be written as the sum of two (not necessarily distinct)
primes [15]. It is worth noting that while the definition of prime numbers
Götz Pfander & Isabel Wünsche
55
is based on multiplication, both – the Twin Prime Conjecture and the Goldbach
conjecture – are stated in terms of the additive structure of integers. This leads
to the question of whether the interplay of multiplication and addition is
the reason that these conjectures, though simple to state, have withstood
centuries of attacks by the most brilliant mathematicians.
The most recent landmark achievement in the theory of prime numbers
is due to Ben Green and Terence Tao (2004) and concerns the additive
structure of prime numbers as well. They proved a classical conjecture
that states that the sequence of prime numbers contains arbitrarily long
arithmetic progressions [16]. That is, for any large number, say 100, there
exists a sequence of 100 prime numbers with the property that the difference
of two successive prime numbers is always the same. The proof of Green
and Tao is certainly not constructive, that is, it does not provide a method
that locates arithmetic progressions in the primes ; the longest arithmetic
progression known today has only 23 terms and starts with
56211383760397, 100758121856257, 145304859952117, ... .
It was found by Markus Frind, Paul Jobling and Paul Underwood in 2004 [17].
The result of Green and Tao described above is only one of a number of
remarkable results which were achieved by collaborations involving Terence
Tao. For this reason, Terence Tao, 31 years old, was awarded a Fields Medal
last year – equivalent to a Nobel Prize for mathematicians.
Mario Merz : Fibonacci numbers as « chiffre for life and art »
Another number sequence that has fascinated scholars and artists alike
is the sequence of Fibonacci numbers [A000045], named after the Italian
mathematician Leonardo of Pisa, also known as Fibonacci. To define them,
one has to fix the numbers 0 and 1 as the first Fibonacci numbers in the
sequence and declare that, however many Fibonacci numbers have been
constructed, the next Fibonacci number is the sum of the two previously
constructed ones, in this case the sum of 0 and 1, that is 1. The next
Fibonacci number is again the sum of the two previous ones, now 1 and 1 ;
their sum equals 2. This procedure defines the obviously infinite Fibonacci
sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ... .
At first glance, the Fibonacci sequence might appear as mystique as the
sequence of primes. The Fibonacci sequence is recursively defined as were
the prime numbers, so to determine the 50th Fibonacci number by the rule
given above, one needs to know the 48th and the 49th Fibonacci numbers. To
compute these, the 46th and 47th Fibonacci numbers are required, and so on.
56
exploring infinity : number sequences in modern art
But contrary to the prime numbers, a simple mathematical analysis shows
that the recursive procedure can be circumvented and Fibonacci numbers
can be computed directly. In other words, the nth Fibonacci number Fn is
given by Binet’s formula
�
� �� � � � �� � � �
�
� ��
�
� � � � � �
�� �
�
This implies that the computation of the 50th Fibonacci number
does not actually require��knowledge
of the 49th, the 48th, the 47th, or the
�
1st Fibonacci �
number. For �instance, when n=50, the formula above gives
12586269025, which is exactly the 50th Fibonacci number.
Above, it was shown that
� sequence of prime numbers not only
� �the
�
�
�
possesses infinitely many terms,
� but has a rich structure which, even
though it has been studied for two thousand years, is still far from being
fully understood. Its structure is so rich that simple statements such as the
existence of infinitely many
�� twin
� primes are still awaiting proof. At the same
� computer power is used nowadays to push the boundary
time, considerable
�
of known prime numbers further and further. In contrast, Fibonacci
numbers have a very simple, well-understood structure as shown by the
formula above. Similar questions
� �� � � � to those asked about the prime numbers
�
are easily answered. For� example,
� not every even number is the sum of
�
two Fibonacci numbers�(12�is not),
and there are not infinitely many twin
Fibonacci numbers. In fact, the authors of this paper are not aware of any
catchy open conjecture involving� the Fibonacci numbers ; furthermore, the
� ��
� � find a Fibonacci number with 10’000’000,
formula above�allows one
to easily
�
�
or even more digits.
� � �
From a mathematical point of view, the infiniteness of Fibonacci
numbers has been resolved and
is exposed. On the other
� ��its �simplicity
� � ��
�
hand, the relation of the Fibonacci
sequence
to the golden ratio and their
�
�
�� �� still
�
� �plays� and is��expected
relevance for growth in nature
to continue to play
�
�
�
�
�
��
� �� � �
a rich role in the visual arts.
�
�
� � �
The Italian Arte Povera artist Mario Merz (*1925) came across the
Fibonacci number sequence in 1967 and made it the basis of his artistic
work. He was particularly attracted by the fact that Fibonacci sequences
appear in biological
settings and can be found in the growth patterns of
�
leaves, snail shells, pine cones, sundry fruits and vegetables, and the skins
of reptiles [18]. The numbers and arrangements of petals, seeds, and plants
that are formed in spirals such as pinecones, pineapples, and sunflowers also
Götz Pfander & Isabel Wünsche
57
adhere to the Fibonacci series. Like the Greek philosopher and geometer
Pythagoras (circa 570 – 490 B. C.) who saw in numbers the ultimate reality,
Merz perceived the Fibonacci numbers as something alive, dynamic and
growing, a « chiffre for life and art ».
In 1968, Merz began to work on his famous igloos, revealing the
prehistoric and tribal features hidden within the present time and space.
These dome-shaped temporary constructions, among them Pythagoras’
Haus of 1994 (illustration 3), express the artist’s preoccupation with the
fundamentals of human existence : shelter, food, and the human relationship
to nature. Each of these archetypal dwellings is built specifically for the
exhibition in which it is shown, the materials – metal tubing, glass, sand bags,
branches, or stone – often being indigenous to the location. The organic
aesthetics of his installations are simultaneously contrasted and enhanced
by the use of neon numbers that refer to the Fibonacci series. Drawing
upon this exponentially growing mathematical sequence, Merz emphasizes
the growth patterns of natural life in his constructions. His usage of the
Fibonacci series as an organizing principle of his works not only suggests
an instance of time as subject matter or a meditation on infinity, but also
intrinsically links natural patterns, human behavior, and artistic creation.
�
� �� � � � � �� � �
� � be the
�
�
� to
The golden ratio was considered by the Greeks
most
� � � � � �
� and has� �
aesthetically pleasing visual proportion
dominated
discourse
on
art
� �
� �� � � � �� � �
�
�
�
�
�
�
�
�
from antiquity
times.
Euclid described the golden ratio in his
� �� � � �to
�� modern
��
� � � � � �
�
�
� ��
�� � books on geometry and number theory, as
Elements,
� �a collection
� � � �of thirteen
�
�
�� �
�� �
� �� � � ��
follows
: « A straight
line is�� ��
said �to�� have
and mean ratio when, as the
�
� � been� cut in extreme
�
� greater
� the
�
� � �so is
whole line is to the greater
segment,
to
the
less
» [19]. The division of
�� �
�� �
�
�
a line
at any point M on the
ratios of lengths AB :
�� AB
�
� line leads to the two
� ��
� which the ratio is equal,
AM �and AM : MB. But there is only one �point M�for
�
�
and the ratio�is then ��
� ��
��
�
�
which�is�� often
called
Φ (capital phi). Its reciprocal
�
�
����� � �
� �� � �
�� � �
� are ��
�
is
(lower case phi).�� Both
these
variously
called the golden
� � of
� �� �
�� �
�
number or golden ratio,
� � �golden �section, golden mean, or the divine
��
�
�� �
�
�
proportion.
�� �
�
�
�
� �� � � �
�
�� �
�
� ratio
The golden
clearly occurs also�� in� Binet’s
formula for the
�
�� � �
�
�
� �� � �
� �
Fibonacci
� �� � � � sequence. Since � � � approaches 0 with increasing n, Binet’s
�
�
�
�
�
��
� ��
� �
�term � �� � � . This explains the
�
�
�
�
formula is actually �dominated
by
the
�
�
�� � �
� � �
�
�
� � Fibonacci
exponential growth���of� the
sequence,
but also implies that the
�
��
�
�
�
�
�
� ��of� �successive
ratio
Fibonacci
numbers
approaches
the golden
ratio, that is,
�
�
�
�� �
�
�
� �� � � � ��
�
�
� � �
��
�
�
� �� � �
�� �
�� �� � � �
�
�
�
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� ��
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�
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�
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�
�
�
�
��
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�
�
�
�
�
� �� � � �
exploring
infinity : number sequences in modern
art
� ��
�
�
��
� �
�
�
�
�� � � �
�
��
� �� �� � �
�
� � � �
�
�
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�
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� � �
�
�
�
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� �� �� �
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� �
58
�
�
�
� ��
�
�
�
�
�� �
�
� �
� �� � � �
�
�
� � �
� �� � � � ��
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�� �� � � �
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�
� �� � �
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� � �
The golden ratio was believed to evoke emotional or aesthetic feelings
and to be particularly pleasing to the human eye [20]. It has been identified
in the construction of the great pyramids in Egypt, the Pyramid of the Sun
�
at Teotihuacán in Mexico, and in Greek temples such as the Parthenon in
Athens [21]. Pythagoras identified the golden ratio in the proportions of the
human body ; his revelations on the proportions of the human figure had a
tremendous impact on Greek and Renaissance art and architecture.
�
In the early 16th century, the golden ratio was rediscovered by Luca
Pacioli (1445 – 1517). In his treatise Divina proportione (On the Divine
Proportion, 1509), he attributes to it five divine attributes. In the first four,
he stated that it is unique like God, is a single proportion in three terms
as the trinity is one substance in three persons, cannot be expressed by a
rational number as god cannot be described in words, and is, like God, always
similar to itself. In the fifth, Pacioli compares the « divine proportion » to the
Platonic quintessence : « God confers being to the celestial virtue, called by the other
name ‘fifth essence’, and through that one to the other four simple bodies, that is, to the
four earthly elements... and so through these to every other thing in nature. Thus this our
proportion is the formal being (according to Timaeus) of heaven, attributing to it the figure
of the solid called Duodecahedron [Dodecahedron], otherwise known as the solid of
twelve pentagons » [22]. Pacioli’s treatise greatly influenced Renaissance artists
such as Leonardo da Vinci, Michelangelo, Raphael, and Albrecht Dürer who
used the golden ratio in their works. In modern times, the golden ratio can
be found in the works of Georges Seurat and Paul Signac as well as Piet
Mondrian and the architect Le Corbusier.
Michael Müller : the mystique of Ulam numbers
Like the Fibonacci number sequence, any Ulam number sequence is
determined by its first two terms [23]. The most famous choice starts with
1 and 2 ; the resulting Ulam sequence is then called (1,2)-Ulam sequence
or, simply, U-sequence [A002858]. The definition of the remaining terms
in an Ulam sequence is again based on a recursive rule. Namely, the next
term of any Ulam sequence is given by the next smallest number that can be
written in exactly one way as the sum of two previously determined Ulam
numbers. Hence, 3=1+2 is an U-number, and so is 4=1+3. The number
5=1+4=2+3 is not a U-number, but 6=2+4 is again a U-number. The Unumber sequence therefore begins with
Götz Pfander & Isabel Wünsche
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1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, 72, … .
Like the Fibonacci sequence, the U-numbers are defined on the basis of
addition only, so it could be expected that U-numbers are not more complex
than the Fibonacci numbers. This is not the case as is described below.
The first question considered here is whether there exist infinitely
many U-numbers. To confirm this, the opposite is assumed, namely that
there are only finitely many U-numbers. In that case, there would be two
largest U-numbers. But their sum is a number that could only be written in
one particular way as the sum of two distinct U-numbers, that is, as the sum
of these two largest ones. This clearly gives rise to another, larger U-number
and thereby contradicts the assumption that made it possible to pick the
largest two U-numbers earlier. The assumption that there are only finitely
many U-numbers must have been wrong, and, therefore, there are infinitely
many U-numbers.
The German artist Michael Müller (*1970) has been fascinated by the
structure of Ulam numbers since the early 1990s. Müller’s interest in the
works of the Polish mathematician Stanislaw Ulam (1909 – 1984) originated
in Ulam’s contributions to war technology as a member of the Manhattan
Project at Los Alamos, New Mexico. There, Ulam invented nuclear pulse
propulsion and, in collaboration with the Hungarian physicist C. J. Everett,
he improved Edward Teller’s early model of the hydrogen bomb. Ulam
was furthermore an early proponent of using computers to perform
mathematical experiments. Looking at Ulam’s many contributions to 20thcentury science and technology, Müller also came across one of Ulam’s
contributions to mathematics, the so-called Ulam numbers [24]. Despite his
growing interest in the U-sequence, Müller did not become aware of the
number theory literature spurred by the creation of the Ulam sequences.
Fascinated by the U-numbers, with their simple definition yet complex
structure, he employed software engineers to implement an algorithm for
the computation of U-numbers on powerful computers. He attempted
to break down the complexity of the Ulam numbers and to increase his
understanding of their structure and density within the natural numbers by
indulging himself in an overwhelming quantity of computed U-numbers.
Like mathematicians before him, he failed to decipher the sequence of Unumbers.
For example, it is known that there are infinitely many U-numbers,
but, as with prime numbers, the obvious question about the density of Unumbers relative to the natural numbers remains to be addressed. In fact,
Stanislaw Ulam conjectured that as with prime numbers, the probability that
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a random choice of an integer would result in a U-number gets smaller
as the chosen number gets larger. This conjecture has not been proven
or disproven, but computations of all U-numbers smaller than 4· 107,
which were carried out by Jud McCranie, seem to imply the opposite : on
average, there are apparently 2 U-numbers in every segment of 27 natural
numbers [25].
The (1,2)-Ulam sequence described above is still hardly understood.
Surprisingly, the complications suddenly vanish when the starting values are
changed. For example, the (2,7)-Ulam sequence [A003668] is given by
2, 7, 9, 11,13, 15, 16, 17,
19, 21, 25, 29, 33, 37, 39, 45, 47, 53, 61, 69, 71, 73, 75, 85, 89, 101,
103, 117, 133, 135, 137, 139, 141, 143, 145, 147, 151, 155, 159, 163,
165, 171, 173, 179, …
and the differences of successive terms are
5, 2, 2, 2, 2, 1, 1,
2, 2, 4, 4, 4, 4, 2, 6, 2, 6, 8, 8, 2, 2, 2, 10, 4, 12, 2, 14, 16, 2, 2, 2, 2, 2,
2, 2, 4, 4, 4, 4, 2, 6, 2, 6, 8, 8, 2, 2, 2, 10, 4, 12, 2, 14, 16, 2, 2, 2, 2, 2,
….
Aside from its first seven terms, the resulting sequence of differences
is apparently periodic and repeats itself every 26 numbers [26, 27], a
phenomenon that is expressed in mathematical terms by saying that the
(2,7)-Ulam sequence is regular [27]. Consequently, the density of regular
Ulam sequences is easily calculated. The list above shows that there are 33
(2,7)-Ulam numbers smaller than 142, and the regularity implies that there
are 59 smaller than 266, 85 smaller than 390, and so on. Therefore, the
density is 33/142, 59/266, 85/390, 111/515, 137/639,… and the fractions
approach the density 13/62. Therefore, about 21% of the integers are
elements of the (2,7)-Ulam sequence. The regularity of this Ulam sequence
also implies that there are no (2,7)-Ulam twins aside of 15 and 16, and 16
and 17. Arbitrarily long arithmetic progressions exist as well, for example
19, 145, 271, 397, 523, 649, 775, 901, 1027, 1153, 1279, 1405, 1531, 1657
is an arithmetic progression of length 15 that is contained in the (2,7)-Ulam
sequence and that can be continued to obtain arithmetic progressions of
any length.
The (2,7)-Ulam sequence has only 2 even terms ; in fact, it has been
shown that any Ulam sequence with finitely many even terms is regular [28].
But even this rather general theorem does not help to decipher the (1,2)Ulam sequence. In fact, among the first 200 elements of the (1,2)-Ulam
sequence there are 105 even terms.
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61
In his attempts to understand the density of the (1,2)-Ulam sequence,
Müller compiled all numbers that are U-numbers and those that are not
into separate books, realizing his Ulam Rot/Grün (Ulam Red/Green, 2005,
illustration 4). Thirteen DIN A4 format books list the numbers from 1 to
226555 in one column per page and one number per line. Whereas a single
green volume includes all U-numbers, the twelve red volumes include all
other natural numbers ; the books represent the inseparability of the natural
numbers.
As with the question on the existence of infinitely many twin primes
that mathematicians and artists such as Rune Mields have explored, Müller
realized that the only successive numbers in his long list of U-numbers are
1,2 ; 2,3 ; 3,4 ; and 47,48. He therefore conjectured that 47 and 48 would be
the largest twin pair of U-numbers. In fact, he devoted two art works to
this phenomenon and announced a prize to those proving or disproving his
assertion. In his 47 48 in 2005 (illustration 5), Müller embossed the numbers
47 and 48 into pink blotting paper and mounted them onto grey cardboard.
The subheadings of the two pieces read accordingly ; they formulate his
assumption that 47 and 48 is the last pair of directly sequential Ulam
numbers. Without knowing, the artist had restated a conjecture that was first
made by Bernardo Recaman in 1973 and which is therefore almost as old as
the U-numbers themselves [29]. The problem is included in the book Open
Problems in Number Theory [30]. Its renowned author, Richard K. Guy, might
have slightly exaggerated the difficulty of solving this problem by stating
that he believes that this problem will « never » be solved [31].
Müller realized that lining up U-numbers in increasing order would
not help him to further understand them, so he decided to develop new
visualization methods that would help to decipher these sequences. His first
method was an adaptation of one of Ulam’s best known contributions to
number theory, namely the Ulam spiral, which Ulam used to obtain new
insights into the sequence of prime numbers [32]. As with U-numbers,
the line up of primes in increasing order appears as if one has just chosen
numbers, that is, odd numbers, at random. The Ulam spiral is obtained
by numbering the squares on a sheet of graph paper, starting from the
center and proceeding in counter-clockwise spirals. The boxes containing
prime numbers are then marked and, surprisingly, the prime numbers
appear not to be distributed on the graph paper at random, but pile up on
specific diagonals. Müller borrowed this procedure, but marked U-numbers
rather than prime numbers on his spiral of natural numbers. As Müller
demonstrates in his 2006 Ulam Spirale (Ulam Spiral, illustration 6), the
resulting picture appears to contain points at random. While noticing the
randomness, Müller also saw the Christian cross appear in the middle of his
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page, adding, in his words « infiniteness of divinity » to the « infiniteness of
the unordered » [33].
Müller’s 2005 work Ulam Ringe (Ulam Rings, illustration 7) presents
another mathematically intriguing visualization of the Ulam number
sequence. The first circles that are drawn have diameters 1 and 2,
corresponding to the first U-numbers. They touch at one point. The next
U-number is 3, so a circle of diameter 3 is required. Since the sum, which
determines the 3, involves the U-numbers 1 and 2, one chooses a circle of
diameter 3, which touches both previously drawn circles again at one point
only. The following U-number is 4 and a circle of diameter 4, touching the
circles with diameter 1 and 3 at one point each, is drawn. The next circle has
diameter 6 and likewise touches circles of diameter 2 and 4, and so on. It is
easy to see that there are at most two circles of a given diameter touching
two previously drawn circles. However, Müller’s experiment leads to the
question, whether the structure of the U-numbers guarantees that circles
corresponding to U-numbers whose sum gives a new U-number are not too
far apart to be touched by a single circle of the prescribed diameter. This
is necessary for the process of drawing circles in the described way to be
continued indefinitely.
Müller’s set of pencil drawings Ulam Gebirgszug (Ulam Mountain Range,
2005, illustration 8) represents a possibility to visualize the Ulam sequence :
the sequence from 260 to 502 merges into a spatial diagram. The shifted
sheets of paper evoke the flow of a river that would culminate, according
to the principle of the sequence, in the number 1 to the right and widen
towards infinity on the left [34].
As for other artists interested in number sequences, Müller’s
fascination with Ulam numbers is more metaphysical than mathematical.
His art represents his own approach to the study of the Ulam numbers.
He did not borrow results from number theory and it is therefore quite
remarkable that his search has led him to the same questions as those of
the number theorists. For Müller, numbers not only serve as a tool to
create structure, but also provide a means to explore infinity. Responding
to the human desire to give order to cosmic chaos, he concentrates in his
work on expressing infinity in finite space. In his quest to visualize infinity
through art, Müller explores the three realms of the human search for
infinity : science, space, and religion. He is interested in mathematics in
order to draw conclusions about infinity ; his works on the U-sequence use
the mathematical systems of a specific number series to visualize infinite
processes from an artistic-scientific viewpoint. Another problem that has
Götz Pfander & Isabel Wünsche
63
informed Müller’s art is the question how to overcome the finiteness of real
space, that is, how to deal with the discrepancy between the finiteness of
the space of human perception and the presumed infiniteness of universal,
cosmic space. His works on star constellations focus on the exploration
of cosmic space in astronomy and philosophy. Religious thought is a third
aspect, which is only indirectly referred to by Müller in his translations of
religious texts into other languages. Thus, Müller’s works are results of
human imagination at the intersection of art and science, philosophy and
religion ; his visual expressions of the infiniteness of number series also
stand for the infiniteness of the universe and the infiniteness of God.
The fascination with number sequences is shared by Roman Opalka,
Rune Mields, Mario Merz, and Michael Müller. Each of these artists chose
a specific number sequence and used it as a means to explore and express
a particular theme going beyond usual human comprehension. Opalka
uses natural numbers to capture the continuous and endless passage of
time. Mields explores chaos and order by means of the complex and only
partially understood system of prime numbers. Merz focuses on growth in
nature, which historically has been associated with the Fibonacci sequence,
and Müller attempts to decipher the U-sequence in his quest to grasp the
infinity of mathematics, space, and divinity. The interest in mathematics per
se distinguishes Mields and Müller from Opalka and Merz. Both, Mields
and Müller, have consulted with mathematicians and their art continues to
evolve as they are drawn deeper and deeper into the uncharted territory of
number theory.
Götz Pfander is with the Mathematics Group, School of Engineering and Science, Jacobs
University Bremen, 28759 Bremen, Germany (e-mail : g.pfander@iu-bremen.de).
Isabel Wünsche is with the Art History Group, School of Humanities and Social Sciences,
Jacobs University Bremen, 28759 Bremen, Germany (e-mail : i.wunsche@iu-bremen.de).
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References
[1] « Zahlen und Zahlenfolgen » (Numbers and Number Sequences), in : Dietmar
Guderian, Mathematik in der Kunst der letzten dreißig Jahre (Mathematics in the art of
the last thirty years), Ebringen Br., 1990, pp. 13 – 22.
[2] Karin von Maur (ed.), Magie der Zahl (The Magic of Numbers), Stuttgart, 1997.
[3] Melvyn B. Nathanson, Elemetary Methods in Number Theory, New York, 2000.
[4] See http ://www.research.att.com/~njas/sequences.
[5] Euclid, Elements, Book IX, Proposition 20, see http :// aleph0.clarku.edu/ ~djoyce/
java/elements/toc.html.
[6] Douglas Stinson, Cryptography : Theory and Practice, 2nd edition, Boca Raton, 2002.
[7] Manindra Agrawal, Neeraj Kayal and Nitin Saxena, « Primes is in P » in : Annals of
Mathematics 160 (2004) 2, pp. 781 – 793. See also http://www.cse.iitk.ac.in/users/
manindra/publications.html.
[8] See http ://www.mersenne.org.
[9] See http ://www.eff.org/awards/coop.php.
[10] Rune Mields, « Vortrag in der Staatlichen Akademie der Bildenden Künste in Stuttgart,
Juli 1994 » (Lecture at the State Academy of the Fine Arts in Stuttgart, July 1994),
in : von Maur, pp. 344 – 347.
[11] Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, New York, 2004.
[12] Rune Mields as cited in : Ulrich Grevsmühl, « Mathematics and Modern Art : Number
Investigations », in : MT 124 September 1988, p. 35.
[13] See note [11].
[14] See http://www.claymath.org/millennium.
[15] See note [11].
[16] Ben Green and Terence Tao, « The primes contain arbitrarily long arithmetic
progressions », see http ://front.math.ucdavis.edu/math.NT/0404188.
[17] The Prime Pp., see http ://primes.utm.edu/glossary/page.php?sort=ArithmeticSeq
uence.
[18] Jay Kappraff, Connections : The Geometric Bridge Between Art and Science, 2nd edition, New
York, 2001.
[19] Euclid, Elements, Book VI, Proposition 30.
[20] See note [18].
[21] For a critical review on the usage of the golden ratio, see : George Markowsky,
« Misconceptions About The Golden Ratio », in : The College Mathematics Journal, 23,
1 (1992), pp. 2 – 19.
[22] Luca Pacioli, De divina proportione (1509), see http ://milan.milanovic.org/math/
english/golden/golden5.html.
[23] Stanislaw Ulam, Problems in Modern Mathematics, New York 1964.
[24] See note [23].
[25] Stephen Wolfram, A New Kind of Science, Champaign, 2002.
[26] James Schmerl and Eugene Spiegel, « On the regularity of some 1-additive sequences »,
in : Journal of Combinatorial Theory, A12, 1994, pp. 172 – 175.
[27] Steven Finch, « Patterns in 1-Addititve Sequences », in : Experimental Mathematics, 1
(1992) 1, pp. 57 – 63.
[28] See note [27].
[29] Bernardo Recaman, « Questions on a sequence of Ulam », in : American Mathematical
Monthly, 80 (1973), pp. 919 – 920.
[30] See note [11].
[31] Personal conversation with the author.
Götz Pfander & Isabel Wünsche
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[32] Myron Stein, Stanislaw Ulam and Mark Wells, « A Visual Display of Some Properties
of the Distribution of Primes», in : American Mathematical Monthly, 71 (1964), pp.
516 – 520 ; Myron Stein and Stanislaw Ulam, « An Observation on the Distribution
of Primes », in : American Mathematical Monthly 74 (1967), pp. 43 – 44.
[33] Personal conversation with the artist.
[34] Clemens Krümmel, « A journey to dust : Michael Müller’s graphic occupations », in :
Michael Müller. 1 m2 ins Schwarz hinein. Teil 1, Klagenfurt 2005, p. 96.
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